| L(s) = 1 | − 5.09e3·2-s + 4.89e4·3-s + 1.75e7·4-s − 2.49e8·6-s + 5.17e9·7-s − 4.68e10·8-s − 9.17e10·9-s + 6.04e11·11-s + 8.61e11·12-s − 7.96e12·13-s − 2.63e13·14-s + 9.13e13·16-s − 1.98e13·17-s + 4.67e14·18-s + 6.27e14·19-s + 2.53e14·21-s − 3.08e15·22-s + 4.55e15·23-s − 2.29e15·24-s + 4.06e16·26-s − 9.10e15·27-s + 9.09e16·28-s + 4.14e16·29-s + 1.35e15·31-s − 7.23e16·32-s + 2.96e16·33-s + 1.01e17·34-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.159·3-s + 2.09·4-s − 0.280·6-s + 0.988·7-s − 1.92·8-s − 0.974·9-s + 0.638·11-s + 0.334·12-s − 1.23·13-s − 1.73·14-s + 1.29·16-s − 0.140·17-s + 1.71·18-s + 1.23·19-s + 0.157·21-s − 1.12·22-s + 0.997·23-s − 0.307·24-s + 2.16·26-s − 0.315·27-s + 2.07·28-s + 0.630·29-s + 0.00959·31-s − 0.354·32-s + 0.101·33-s + 0.247·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.9114543267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9114543267\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 5.09e3T + 8.38e6T^{2} \) |
| 3 | \( 1 - 4.89e4T + 9.41e10T^{2} \) |
| 7 | \( 1 - 5.17e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 6.04e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 7.96e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.98e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.27e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 4.55e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 4.14e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.35e15T + 2.00e34T^{2} \) |
| 37 | \( 1 + 3.41e17T + 1.17e36T^{2} \) |
| 41 | \( 1 + 3.69e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 1.96e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.44e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 6.39e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 2.81e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 4.67e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 2.77e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.29e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 4.56e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 3.99e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.45e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 1.80e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 8.25e22T + 4.96e45T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.90193002020823932339913474313, −11.19915736248966531597929686851, −9.855702876871078000352675485762, −8.849073239713488833418643593286, −7.921785982507148123834097495642, −6.86484226012888738802830151210, −5.11990928209704657605271777585, −2.90583598968921865551321300499, −1.73365832574989764201565941153, −0.62933366023716082329223072198,
0.62933366023716082329223072198, 1.73365832574989764201565941153, 2.90583598968921865551321300499, 5.11990928209704657605271777585, 6.86484226012888738802830151210, 7.921785982507148123834097495642, 8.849073239713488833418643593286, 9.855702876871078000352675485762, 11.19915736248966531597929686851, 11.90193002020823932339913474313
